Cation ordering in BaAl2Ge2O8-feldspar: implications for the phase transition in anorthite

BaAl₂Ge₂O₈-Feldspar undergoes an order-disorder phase transition I2/c↔C2/m at T _tr ≈1690 K. The thermodynamics of the Al,Ge cation ordering process is described in terms of the compressible Ising model in mean field approximation. The mean field potential predicts a first order character of the phase transition. This is compared to antiferromagnetic ordering in a two-dimensional square Ising model with NN-pair interactions and four-spin interactions on alternating squares. Calculated order parameters and short range ordering are in good agreement with the corresponding properties observed in BaAl₂Ge₂O₈-feldspar by means of X-ray diffraction, hard mode infrared spectroscopy and TEM. Using known calorimetric data a similar model is postulated for Al,Si ordering in anorthite, CaAl₂Si₂O₈, for which the derived potential describes a transition with slightly stronger first order character at T _tr ≈1928 K. Received: 30 January 1998 / Revised, accepted: 29 August 1998     

Determination of the mechanism of cation ordering in magnesioferrite (MgFe2O4) from the time- and temperature-dependence of magnetic susceptibility

The kinetics of non-convergent cation ordering in MgFe₂O₄ have been studied by measuring the Curie temperature (T _c) of synthetic samples as a function of isothermal annealing time. The starting material was a synthetic sample of near-stoichiometric MgFe₂O₄, synthesised from the oxides in air and quenched from 900 °C in water. Ordering experiments were performed using small chips of this material and annealing them at temperatures between 450 °C and 600 °C. The chips were periodically removed from the furnace, and their Curie temperatures were determined from measurements of alternating-field magnetic susceptibility (χ) as a function of temperature (T) to 400 °C. The Curie temperature of MgFe₂O₄ is very sensitive to the intracrystalline distribution of Fe³⁺ and Mg cations between tetrahedral and octahedral sites of the spinel crystal structure, and hence provides a very sensitive probe of the cation ordering process. The χ-T curve for the starting material displays a single sharp magnetic transition at a temperature of 303 °C. During isothermal annealing, the χ-T curve develops two distinct magnetic transitions; the first at a temperature corresponding to T _c for the disordered starting material and the second at a higher temperature corresponding to T _c for the equilibrium ordered phase. The size of the magnetic signal from the ordered phase increases smoothly as a function of time, until equilibrium is approached and the shape of the χ-T curve corresponds to a single sharp magnetic transition for the homogeneous ordered phase. These observations demonstrate that cation ordering in MgFe₂O₄ proceeds via a heterogeneous mechanism, involving the nucleation and growth of fine-scale domains of the ordered phase within a matrix of disordered material. Disordering experiments were performed by taking material equilibrated at 558 °C and annealing it at 695 °C. The mechanism of isothermal disordering is shown to involve nucleation and growth of disordered domains within an ordered matrix, combined with continuous disordering of the ordered matrix. This mixed mechanism of disordering may provide an explanation for the difference between the rates of ordering and disordering observed in MgFe₂O₄ using X-ray diffraction. The origin of the heterogeneous ordering/disordering mechanism is discussed in terms of the Ginzburg-Landau rate law. It is argued that heterogeneous mechanisms are likely to occur in kinetic experiments performed far from equilibrium, whereas a homogeneous mechanism may operate under slow equilibrium cooling. The implications of these observations for geospeedometry are discussed. Received: 12 May 1998 / Accepted: 25 June 1998     

Cation ordering of orthopyroxenes from the Skaergaard Intrusion: implications for the subsolidus cooling rates and permeabilities

 We have determined the quenched cation ordering states of five orthopyroxene crystals collected from the marginal border group and the lower zone a and b of the Skaergaard intrusion, and modeled these data to retrieve their closure temperatures (T _c) of Fe–Mg ordering and cooling rates. According to existing thermal models for the Skaergaard pluton, conductive cooling dominated the high and low temperature regimes, which were separated by an intermediate temperature regime in which the cooling was controlled primarily by convective fluid circulation. The cooling rates retrieved from the quenched cation ordering states of the orthopyroxene crystals strictly apply to temperatures around the closure temperatures of the ordering states, ∼340–400^° C, which fall at the transition from convective to the lower temperature conductive cooling. The cooling rates obtained from the cation ordering states of orthopyroxene vary from ∼1 to 270 K/ka. These results agree well with a thermal model calculated using an assumed average permeability of 10^-12 cm² for the pluton, but not completely with a model calculated on the basis of an average permeability of 10^-13 cm², although both values produced shifts of δ¹⁸O that are comparable to those observed in the pluton. Received: 27 February 1995/Accepted: 20 July 1995     

Computational study of tetrahedral Al–Si ordering in muscovite

 The nature of Al–Si ordering across the tetrahedral sites in muscovite, K₂Al₄(Si₆Al₂O₂₀)(OH)₄, was investigated using various computational techniques. Values of the atomic exchange interaction parameters J _l were obtained. From these parameters, a two-dimensional Al–Si ordering scheme was deduced. The transition temperature T _c for this two-dimensional ordering is 1900 K. There are several possible ordering schemes in three dimensions, based on different stacking sequences of ordered sheets of tetrahedral sites. Monte Carlo simulations of both two-dimensional and three-dimensional ordering were performed, but in the three-dimensional simulation only the two-dimensional ordering is seen, implying that three-dimensional ordering is too slow to be attained during the timescale of the simulation. The effect of the three-dimensional interactions is to raise the two-dimensional ordering temperature to 2140 K. From the three-dimensional Monte Carlo simulation, the frequency of occurrence of 4Si0Al, 3Si1Al, 2Si2Al and 1Si3Al clusters was determined, which match those inferred by ²⁹Si MAS–NMR measurements reasonably well. In fact, the match suggests that the cation ordering seen in experiments corresponds to a configuration with considerable short-range order but no long-range order, similar to a state that is at a temperature just above an ordering phase transition. Received: 28 August 2000 / Accepted: 12 March 2001     

Octahedral cation ordering in olivine at high temperature. II: an in situ neutron powder diffraction study on synthetic MgFeSiO4 (Fa50)

 The partitioning of Fe and Mg between the M1 and M2 octahedral sites of olivine has been investigated by in situ time-of-flight neutron powder diffraction. The degree of M-cation order was determined from direct measurements of site occupancies in a synthetic sample of Fo50Fa50 heated to 1250 °C at the Fe-FeO oxygen buffer. Fe shows slight preference for M1 at temperatures below about 600 °C, progressively disordering on heating to this temperature. Above 630 °C, the temperature at which site preferences cross over (T _cr), Fe preferentially occupies M2, becoming progressively more ordered into M2 on increasing temperature. The cation-ordering behaviour is discussed in relation to the temperature dependence of the M1 and M2 site geometries, and it is suggested that vibrational entropy, crystal field effects and changes in bond characteristics play a part in the cross-over of partitioning behaviour. The temperature dependence of site ordering is modelled using a Landau expansion of the free energy of ordering of the type ΔG = −hQ + gTQ +  (T − T _c)Q ² +  Q ⁴, with a/h = 0.00406 K⁻¹, b/h = 2.3, T _c = 572 K and g/h = 0.00106 K⁻¹. These results suggest that the high-temperature ordering behaviour across the forsterite-fayalite join will have a bearing on the activity-composition relations of this important rock-forming mineral, and indicate that Fe-Mg olivine solid solutions become less ideal as temperature increases. Received: 12 August 1999 / Accepted: 25 April 2000     

Thermal expansion and high-temperature P21/c–C2/c phase transition in clinopyroxene-type LiFeGe2O6 and comparison to NaFe(Si,Ge)2O6

A synthetic clinopyroxene with composition LiFe³⁺Ge₂O₆, monoclinic s.g. P2₁/c, a = 9.8792(7), b = 8.8095(5), c = 5.3754(3) Å, β = 108.844(6)°, V = 442.75(16) ų, has been studied by in situ low- and high-temperature single-crystal X-ray diffraction. The variation of lattice parameters and the intensity of the b-type reflections (h + k = 2n + 1, only present in the P-symmetry) with increasing temperature showed a displacive phase transition from space group P2₁/c to C2/c at a transition temperature T _tr = 789 K, first order in character, with a sudden volume increase of 1.6% and a decrease of β by 1° at the transition. This spontaneous dilatation is reversible, shows a limited hysteresis of ±10°C, and corresponds to the vanishing of the b-type reflections, thus indicating a symmetry increase to space group C2/c. Below T _tr an expansion is observed for all the cell parameters, while the β angle remained almost constant; at T > T _tr the thermal volume expansion is due to dilatation of the structure in the $(\bar{1}\,0\,1) A synthetic clinopyroxene with composition LiFe³⁺Ge₂O₆, monoclinic s.g. P2₁/c, a = 9.8792(7), b = 8.8095(5), c = 5.3754(3) ?, β = 108.844(6)°, V = 442.75(16) ?³, has been studied by in situ low- and high-temperature single-crystal X-ray diffraction. The variation of lattice parameters and the intensity of the b-type reflections (h + k = 2n + 1, only present in the P-symmetry) with increasing temperature showed a displacive phase transition from space group P2₁/c to C2/c at a transition temperature T _tr = 789 K, first order in character, with a sudden volume increase of 1.6% and a decrease of β by 1° at the transition. This spontaneous dilatation is reversible, shows a limited hysteresis of ±10°C, and corresponds to the vanishing of the b-type reflections, thus indicating a symmetry increase to space group C2/c. Below T _tr an expansion is observed for all the cell parameters, while the β angle remained almost constant; at T > T _tr the thermal volume expansion is due to dilatation of the structure in the ([`1] 0 1)(\bar{1}\,0\,1) plane, mostly along [0 1 0], and pure shear in the (0 1 0) plane due to the decrease of β. From comparison with silicate analogues, the germanate clinopyroxenes are more expansible, while the P2₁/c expands more than the C2/c phase. The evolution of Q ² (calculated as the normalized intensity of b-type reflections) with T in the framework of the Landau theory has been done using a standard expression for a first order phase transition. We observe a jump of Q ₀² = 0.538(2) at T _tr, with T _c of 481(7) K, b/a = −2,290 K, and c/a = 3,192 K, and thus far from being tri-critical point. A closely related composition (LiFe³⁺Si₂O₆) shows an equivalent phase transition at 228 K, which is very close to the tri-critical point and 561 K cooler. This result indicates that a change in the composition of tetrahedral sites can have dramatic effects on the P2₁/c ↔ C2/c displacive phase transition in clinopyroxenes. The major changes observed in the evolution of the crystal structure with T are observed in the M2 polyhedron, with a volume decrease by ca. 13.3%, compared to ca. 1.3% observed in the M1 polyhedron. The tetrahedra behave as rigid units with neither a significant change of volume at T > T _tr (<1‰), nor a change of tilting of the basal plane. No change in coordination is observed at T > T _tr in the M2 polyhedron, which remains sixfold coordinated although a strong deformation of this polyhedron is observed. This deformation is related to a strong change by 51.4° at T _tr of the kinking angle (O3–O3–O3 angle) of the B-chain of tetrahedra, which switches from O-rotated to S-rotated [from 143.3(5)° to 194.7(6)°]. The A-chain is S-rotated at T < T _tr [206.8(5)° at 703 K] and extends by 12° at the transition.     

Kinetics of Fe<Superscript>2+</Superscript>–Mg order–disorder in orthopyroxene: experimental studies and applications to cooling rates of rocks

We determined the forward rate constant (K⁺) for the Fe²⁺–Mg order–disorder between the M2 and M1 sites of orthopyroxene (OPx), which is described by the homogeneous reaction Fe²⁺ (M2) + Mg(M1) ↔ Mg(M2) + Fe²⁺ (M1), by both ordering and disordering experiments at isothermal condition and also by continuous cooling experiments. The rate constant was determined as a function of temperature in the range of 550–750°C, oxygen fugacity between quartz–fayalite–iron and Ni–NiO buffers, and at compositions of 16 and 50 mol% ferrosilite component. The K⁺ value derived from disordering experiment was found to be larger than that derived from ordering experiment at 550°C, while at T>580°C, these two values are essentially the same. The fO₂ dependence of the rate constant can be described by the relation K⁺ α (fO₂) ⁿ with n=5.5–6.5, which is compatible with the theoretically expected relation. The Arrhenius relation at the WI buffer condition is given by

Al-Si ordering in Sr-feldspar SrAl2Si2O8: IR,TEM and single-crystal XRD evidences

Al-Si ordering in Sr-feldspar has been followed by isothermal annealing, starting from a disordered metastable configuration. Ordering could not be followed by changes in the spontaneous strain as cell parameters did not show significant changes with thermal treatment from 0.016 h to 452 h at T=1350° C, while, on the contrary, significant changes in IR spectra are observed. A single crystal obtained from melt (Q _od 0) has been progressively heated up to 678 h at T=1350° C and the relevant structural refinements enabled to monitor changes in degree of Al-Si order up to Q_od = 0.86. In isothermal treatment for Sr-feldspar it is observed a significantly lower Q _od than in anorthite after the same annealing time. TEM observation has shown in Sr-feldspar, also for shortest annealing, b type reflections, while in anorthite, in the same conditions, e type reflections have been observed (Carpenter 1991a). In the first stages of ordering b APDs sized 100 Å (at T=1350° C, 0.33 h) have been observed in Sr-feldspar; APD coarsening occurs with an activation energy of 120±7 kcal mol^-1, not significantly different from anorthite. The ordering process seems to be a slower process in Sr-feldspar than in anorthite, even though data from longer annealing suggest that the Q _od close to the equilibrium is the same in Sr and Ca-feldspar (Q _od = 0.86 at T=1350° C).     

H T -XRD study of synthetic ferrian magnesian spodumene: the effect of site dimension on the P 21/ c → C 2/ c phase transition

 Ferrian magnesian spodumene was synthesized in the MLFSH system at P=0.4 GPa, T=700 °C, fO₂=NNO+2.3. The space group at room T is P2₁/c [a=9.638(3) ?, b=8.709(2) ?, c=5.258(2) ?, β=109.83(3)^∘, V=415.2 ?³]. The structure is topologically equivalent to that of ferrian spodumene, LiFeSi₂O₆, and has two symmetrically independent tetrahedral chains, A and B, and two independent octahedral sites, M1 and M2. The crystal-chemical composition was determined combining EMP, SIMS and single-crystal XRD analysis, yielding ^M2(Li_0.85Mg_0.09Fe²⁺ _0.06) ^M1(Fe³⁺ _0.85Mg_0.15)Si₂O₆. Li is ordered at the M2 site and Fe³⁺ is ordered at the M1 site, whereas Mg (and Fe²⁺) distribute over both octahedral sites. Structure refinements done at different temperatures (25, 70, 95, 125, 150 and 200 °C) allowed characterization of a reversible displacive P2₁/c→C2/c transition at 106 °C. Previous HT-XRD studies of Li-clinopyroxenes had shown that the transition temperature is inversely related to the size of the M1 cation. For the crystal of this work, the aggregate ionic radius at M1 is longer than that of ferrian spodumene, for which the transition temperature is −44 °C. The higher transition temperature observed can only be explained on the basis of the shorter aggregate radius at the M2 site (due to the presence of Mg substituting after Li), in keeping with the results obtained for ferromagnesian P2₁/c pyroxenes. The effects of all the chemical substitutions must be considered when modelling transition temperatures and thermodynamic behaviour in clinopyroxenes. Received: 7 May 2002 / Accepted: 23 October 2002     

A statistical-mechanics model of ordering in aluminosilicate solid solutions

A statistical-thermodynamic approximation for order-disorder phase transitions in aluminosilicate solid solutions is presented. The approximation involves estimating the number of configurations with long range order parameter Q and short range order parameter σ, using an approximation to the probability that a configuration with the correct amount of long range order will also have the correct amount of short range order. This estimate is then used to give a free energy F(Q, σ, x, T), where x is the concentration of Al in the structure, and hence quantities such as T _c (x) are estimated. The predictions of the model for T _c (x) and the critical concentration x _c at which T _c falls to zero are shown to be in good agreement with the results of Monte Carlo simulations. Received: 14 May 1997 / Revised, accepted: 2 June 1997     

Time-temperature-transformation (TTT) analysis of cation disordering in omphacite

The kinetics of cation disordering in a natural ordered (P2/n) omphacite have been followed at P=18 and 30 kb, T= 750–1,260° C, for times of between 1.5 min and 16 days in a piston-cylinder apparatus. Time-temperature-transformation (TTT) analysis of the experimental data, using the presence or absence of the 11¯1 reflection in single crystal X-ray precession photographs to indicate the extent of reaction, yields an equilibrium order/disorder temperature (T _ord) of 865±10° C, an activation enthalpy (1 bar) of 71±6 kcal mole^–1 and an activation volume of 9±4 cm³ mole^–1 (plus and minus figures represent the precision of a best fit between experimental data and TTT theory rather than absolute errors). The activation volume is consistent with a vacancy mechanism of cation diffusion. H₂O, added in the form of oxalic acid, appears to speed the process up slightly. The overall transformation mechanism is continuous, involving neither the nucleation of a disordered phase nor a change in antiphase-domain distribution. This is consistent with both first- and non-first-order character for the C2/cP2/n transformation, though a range of ordered states below T _ord is indicated by the weakening of h+k=odd reflections. A simple extrapolation of the disordering rates to geological conditions leads to the first estimate of how long disordered omphacites would take to order in nature, ranging from less than one year at T800° C to more than 10⁷ years at T<350° C.     

Thermodynamics of Al/Al avoidance in the ordering of Al/Si tetrahedral framework structures

The main driving force behind Al/Si ordering in tetrahedral framework aluminosilicates is nearest-neighbour Al/Al avoidance. Computer simulation is used to explore the direct consequences of such Al/Al avoidance. The main result is that the order-disorder transition temperature T _c falls dramatically as the concentration x of Al in the structure is reduced, and if the only interactions are those associated with nearest-neighbour Al/Al avoidance, T _c becomes zero for x less than some critical value x _c , where x _c =0.31 for the feldspar framework and x _c =0.34 for cordierite. Also a large degree of short range order is found above T _c . Both results differ radically from the standard Bragg-Williams model. Plots of entropy and enthalpy of ordering are given as functions of x and T, which may be used to interpret experimental data or for extrapolation into ranges of x and T inaccessible to experiment. Received: 14 May 1997 / Revised, accepted: 2 June 1997     

High-T phase transition of synthetic ANaB(LiMg)CMg5Si8O22(OH)2 amphibole: an X-ray synchrotron powder diffraction and FTIR spectroscopic study

The synthetic amphibole Na_0.95(Li_0.95Mg_1.05)Mg₅Si₈O₂₂(OH)₂ was studied in situ at high-T, using IR OH-stretching spectroscopy and synchrotron X-ray powder diffraction. At room-T the sample has P2₁ /m symmetry, as shown by the FTIR spectrum. It shows in the OH region two well-defined and intense absorptions at 3,748 and 3,712 cm⁻¹, respectively, and two minor bands at 3,667 and 3,687 cm⁻¹. The main bands are assigned to the two independent O–H groups in the primitive structure. The two minor bands evidencing the presence of small amount of vacant A-site (^A□_0.05). With increasing T, these bands shift continuously and merge into a unique absorption at high temperature. A change as a function of increasing T is revealed by the evolution of the refined unit-cell parameters, whose trend shows a transition to C2/m at about 320–330°C. The spontaneous scalar strain, fitted with a tricritical 2–6 Landau potential, gives a T _c of 325(10)°C (β parameter = 0.27). Comparison with the second-order P2₁ /m ⇔ C2/m phase transition at 255°C for synthetic amphibole ^ANa_0.8^B(Na_0.8Mg_1.2)^CMg₅Si₈O₂₂(OH)₂ indicates that the substitution of Na with Li at the B-sites strongly affects the thermodynamic character and the T _c of the phase transition. The comparison of LNMSH amphiboles with cummingtonitic ones shows that the high-T thermodynamic behaviour is affected by A-site occupancy.     

LiFeSi<Subscript>2</Subscript>O<Subscript>6</Subscript> and NaFeSi<Subscript>2</Subscript>O<Subscript>6</Subscript> at low temperatures: an infrared spectroscopic study

 Synthetic aegirine LiFeSi₂O₆ and NaFeSi₂O₆ were characterized using infrared spectroscopy in the frequency range 50–2000 cm⁻¹, and at temperatures between 20 and 300 K. For the C2/c phase of LiFeSi₂O₆, 25 of the 27 predicted infrared bands and 26 of 30 predicted Raman bands are recorded at room temperature. NaFeSi₂O₆ (with symmetry C2/c) shows 25 infrared and 26 Raman bands. On cooling, the C2/c–P2₁/c structural phase transition of LiFeSi₂O₆ is characterized by the appearance of 13 additional recorded peaks. This observation indicates the enlargement of the unit cell at the transition point. The appearance of an extra band near 688 cm⁻¹ in the monoclinic P2₁/c phase, which is due to the Si–O–Si vibration in the Si₂O₆ chains, indicates that there are two non-equivalent Si sites with different Si–O bond lengths. Most significant spectral changes appear in the far-infrared region, where Li–O and Fe–O vibrations are mainly located. Infrared bands between 300 and 330 cm⁻¹ show unusually dramatic changes at temperatures far below the transition. Compared with the infrared data of NaFeSi₂O₆ measured at low temperatures, the change in LiFeSi₂O₆ is interpreted as the consequence of mode crossing in the frequency region. A generalized Landau theory was used to analyze the order parameter of the C2/c–P2₁/c phase transition, and the results suggest that the transition is close to tricritical. Received: 21 January 2002 / Accepted: 22 July 2002     

Mode of formation of hibonite (CaAl12O19) within the U-Th skarns from the granulites of S-E Madagascar

 In Madagascar, hibonite occurs as a rather frequent mineral within thorianite-bearing skarns which are widespread in the Pan African granulitic formations constituting the S-E part of the Island (Tranomaro area). In these skarns, leucocratic segregations made up of CO₃-scapolite to meionite (An_equivalent=89–95% which implies T≥850° C), spinel and corundum were formed at stage 1 of metasomatism in a titanite-bearing matrix consisting of scapolite (An_eq=77–88) and aluminous diopside. During stage 2 of metasomatism, scapolite from the lenses were altered to anorthite+calcite while the less calcic scapolite remained stable which indicates T≈800° C. Hibonite crystallized at the expense of corundum and spinel. Expressed as mol% of the CaAl₁₂O₁₉/Ca(Al₁₀TiR²⁺)O₁₉/REE(Al₁₁R²⁺)O₁₉ [+Th (Al₁₀R²⁺ ₂)O₁₉] end-members (R ²⁺=Mg, Fe²⁺, Zn²⁺; Al=Al, Fe³⁺; Ti=Ti, Si), its composition varies from 26/72/2 to 50/23/27. The ideal activity of the CaAl₁₂O₁₉ component is about 0.25. Fluid inclusions in corundum, hibonite and anorthite are composed of nearly pure CO₂. In corundum, the isochores for primary inclusions are in agreement with the P-T estimates for regional metamorphism and stage 1 metasomatism (T≈850° C, P≈5 kbar). Inclusions with the highest density in hibonite and anorthite constrain P to about 3–3.5 kbar for T=800° C. Thermodynamic calculations indicate that, in addition to a low activity of CaAl₁₂O₁₉, stability of hibonite in equilibrium with anorthite and calcite implies an extremely low activity of silica (below the zircon-baddeleyite buffer). By contrast the activity of CO₂ may be high, in agreement with the observed fluid compositions. These results are corroborated by a short comparison with the other granulite occurrences of hibonite in Tanzania and South India. Received: 18 August 1994 / Accepted: 12 October 1995     

Structural thermal behavior of paragonite and its dehydroxylate: a high-temperature single-crystal study

 The lattice constants of paragonite-2M₁, NaAl₂(AlSi₃)O₁₀(OH)₂, were determined to 800 °C by the single-crystal diffraction method. Mean thermal expansion coefficients, in the range 25–600 °C, were: α_a = 1.51(8) × 10⁻⁵, α_b = 1.94(6) × 10⁻⁵, α_c = 2.15(7) ×  10⁻⁵ °C⁻¹, and α_V = 5.9(2) × 10⁻⁵ °C⁻¹. At T higher than 600 °C, cell parameters showed a change in expansion rate due to a dehydroxylation process. The structural refinements of natural paragonite, carried out at 25, 210, 450 and 600 °C, before dehydroxylation, showed that the larger thermal expansion along the c parameter was mainly due to interlayer thickness dilatation. In the 25–600 °C range, Si,Al tetrahedra remained quite unchanged, whereas the other polyhedra expanded linearly with expansion rate proportional to their volume. The polyhedron around the interlayer cation Na became more regular with temperature. Tetrahedral rotation angle α changed from 16.2 to 12.9°. The structure of the new phase, nominally NaAl₂ (AlSi₃)O₁₁, obtained as a consequence of dehydroxylation, had a cell volume 4.2% larger than that of paragonite. It was refined at room temperature and its expansion coefficients determined in the range 25–800 °C. The most significant structural difference from paragonite was the presence of Al in fivefold coordination, according to a distorted trigonal bipyramid. Results confirm the structural effects of the dehydration mechanism of micas and dioctahedral 2:1 layer silicates. By combining thermal expansion and compressibility data, the following approximate equation of state in the PTV space was obtained for paragonite: V/V ₀ = 1 + 5.9(2) × 10⁻⁵ T(°C) − 0.00153(4) P(kbar). Received: 12 July 1999 / Revised, accepted: 7 December 1999     

Cation ordering in NiAl2O4 spinel by a 111 systematic row CBED technique

A focussed probe wide angle systematic row CBED technique has been used to determine the degree of cation inversion, i.e. the magnitude of the cation ordering parameter x, in (Ni_{1− x} Al _x ) [Ni _{x /2}Al_{1− x /2}]₂O₄ spinel to an estimated accuracy of ±0.1. For comparison purposes, the same technique has also been applied to an MgCr₂O₄ spinel (where very little cation inversion is expected). The simplicity of this CBED technique in conjunction with the fact that the experimental data can be obtained from small illuminated areas several tens to 100 nm in diameter, suggests that it may be a very useful technique for estimating the extent of cation inversion in multi-phase mineralogical specimens containing spinels. Received: 19 October 1998 / Revised, accepted: 14 June 1999     

A computational study of Al/Si ordering in cordierite

The ordering of Al and Si in Mg cordierite Mg₂Al₄Si₅O₁₈ is considered using computer simulation. First the enthalpy of interaction J _ij between sites is derived by computer modelling 101 different Al/Si configurations and analysing their energies. They are compared with similar results for three other minerals and with ab initio calculations to assess the whole approach. Secondly the ordering process is studied using Monte Carlo simulation applied to the J _ij . The ordering phase transition temperature T _c is found as 1800°C in reasonable agreement with the experimental estimate of 1450° C. These are much lower than the estimate T _c(ABW)≈7600°C obtained from Bragg-Williams theory. Strong short-range order sets in below T _c(ABW), and the reasons for much lower temperature T _c of long-range ordering are discussed. Strong short-range also sets in very rapidly in a simulated anneal, in agreement with experiment. Thirdly an attempt is made to compare our calculated enthalpies directly with the results of NMR and calorimetry experiments, not completely successfully. A free energy ΔG≈4.6 eV for the activation barrier for ordering is suggested.     

Empirical constraints on closure temperatures from a single diffusion coefficient

The elucidation of thermal histories by geochronological and isotopic means is based fundamentally on solid-state diffusion and the concept of closure temperatures. Because diffusion is thermally activated, an analytical solution of the closure temperature (T _c ^*) can only be obtained if the diffusion coefficient D of the diffusion process is measured at two or more different temperatures. If the diffusion coefficient is known at only one temperature, however, the true closure temperature (T _c ^*) cannot be calculated analytically because there exist an infinite number of possible (apparent) closure temperatures (Tˉ _c ) which can be generated by this single datum. By introducing further empirical constraints to limit the range of possible closure temperatures, however, mathematical analysis of a modified form of the closure temperature equation shows that it is possible to make both qualitative and quantitative estimates of T _c ^* given knowledge of only one diffusion coefficient D _M measured at one temperature T _M . Qualitative constraints of the true closure temperature T _c ^* are obtained from the shapes of curves on a graph of the apparent T _c (Tˉ _c ) vs. activation energy E, in which each curve is based on a single diffusion coefficient measurement D _M at temperature T _M . Using a realistic range of E, the concavity of the curve shows whether T _M is less than, approximately equal to, or greater than T _c ^*. Quantitative estimates are obtained by considering two dimensionless parameters [ln êRT^ _c vs. T _c ^*/T _M ] derived from these curves. When these parameters are plotted for known argon diffusion data and for a given diffusion size and cooling rate, it is found that the resultant curves are almost identical for all of the commonly dated K–Ar minerals – biotite, phlogopite, muscovite, hornblende and orthoclase – in spite of differences in their diffusion parameters. A common curve for Ar diffusion can be derived by least-squares fitting of all the Ar diffusion data and provides a way of predicting a “model” closure temperature T _cm from a single diffusion coefficient D _M at temperature T _M . Preliminary diffusion data for a labradorite lead to a T _cm of 507 ± 17 °C and a corresponding activation energy of about 65 kcal/mol, given a grain size of 200 μm and a cooling rate of 5 °C/Ma. Curves for He diffusion in silicates (augite, quartz and sanidine) also overlap to a significant degree, both among themselves and with the Ar model curve, suggesting that a single model curve may be a good representation of noble gas closure temperatures in silicates. An analogous model curve for a selection of ¹⁸O data can also be constructed, but this curve differs from the Ar model curve. A single model curve for cationic species does not appear to exist, however, suggesting that chemical bonding relationships between the ionic size/charge and crystal structure may influence the closure temperatures of diffusing cations. An indication of the degree of overlap among the various curves for Ar, He, ¹⁸O and cations is also obtained by considering the dimensionless parameter E/RT _c ^*; for the noble gases and ¹⁸O, E/RT _c ^* values for the respective minerals are very similar, whereas for cations, there is significant dispersion. Given these constraints, this may be a potential method of estimating closure temperatures for certain diffusing species when there are limited diffusion data. Received: 1 July 1999 / Accepted: 24 March 2000